University of Cincinnati

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1 University of Cincinnati Date: 2/7/2011 I, Abir K Sengupta, hereby submit this original work as part of the requirements for the degree of Master of Science in Civil Engineering. It is entitled: Instrumentation and Load Rating of Steel Curved Girder Bridges Student's name: Abir K Sengupta This work and its defense approved by: Committee chair: James Swanson, PhD Committee member: Arthur Helmicki, PhD Committee member: Victor Hunt, PhD Committee member: Gian Rassati, PhD 1363 Last Printed:2/18/2011 Document Of Defense Form

2 Instrumentation and Load Rating of Steel Curved Girder Bridges A thesis submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of Master of Science in the Department of Civil and Environmental Engineering of the College of Engineering and Applied Science by Abir Sengupta B. Tech College of Engineering, Pune June 2007 Committee Chair: James Swanson, Ph.D.

3 Abstract Curved girder bridges are frequently used by state departments of transportation because they add significant flexibility in the determination of highway alignments, especially at congested interchanges. Much of the seminal research into the behavior of curved girder bridges was conducted in the late 1960s. The development of computer programs for the analysis and design of curved-girder bridges quickly followed and they have become more advanced over time. While the load path for a straight girder bridge is such that the girders are subjected primarily to shear forces and strong-axis bending moments, the load path in curved girders bridges inherently includes eccentric loads that result not only in strong axis shear forces and bending moments but also torsional moments, warping, and the resulting shear. As a result, the level of complexity in modeling a curved bridge is increased exponentially when compared to a straight bridge. A secondary consideration is the selection of girder configurations for curved-girder bridges. While I-shaped girders are often selected as the de facto section of choice for straight bridges, box girders offer significant advantages over I-shaped girders in curved bridges because of the relatively high torsional rigidity that they offer. This article addresses the most significant issues involved with the analysis and design of curved girder bridges starting with a review of the mechanics associated with torsion, which acts on the members, moving into a review of research conducted to date, which is then followed by a summary of design provisions. Next, the topic of finite element modeling of curved girder bridges in 2D and 3D will be addressed. Finally, a discussion of evaluation and load rating of curved bridges will be presented. As a final consideration, a study comparing the behavior of tangent bridges (bridges made up of straight girders supporting a horizontally curved roadway) with the behavior of curved bridges will be conducted and included in the thesis. ii

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5 Table of contents List of Figures vi List of Tables...ix Chapter 1-Torsion Introduction Shear Center Basics of Torsion Torsion in Solid Circular Sections Torsion in Rectangular Sections Thin walled Hollow Torsional Members Torsion in Open Sections Torsion Example...16 Chapter 2-Bridges Curved in Plan Introduction Literature Survey Economy and Aesthetics Methods of Analysis Example Problem Chapter 3-Comparision between Open and Closed Box Curved Girder Bridge Introduction Mechanics...34 iv

6 3.3 Fabrication and Erection Economy...37 Chapter 4-Tangent Bridges Introduction Mechanics LRFD Specifications-Curved vs Straight Summary...44 Chapter 5-Stress Distribution and Instrumentation Introduction Cantilever Model (Straight Girder) Cantilever Model (Curved Girder) Instrumentation...52 Chapter 6- Instrumentation of a Curved Girder Bridge Introduction Bowing Action Stress Distribution-Method I Plots and Conclusions Stress Distribution-Method II Future Work...82 Chapter 7 - Load Rating of a Curved Girder...83 v

7 7.1 Introduction Allowable Stress Load Rating Load Rating for Shear, Moment and Torsion Load Rating for all three stresses at a cross-section of I-girder Principal Stresses Summary- Load Rating Appendix A-Example: Design of Curved Stringer Bridge...94 Appendix B-Recalculation of Shear, Moment and Torsion for a Straight Beam Appendix C-Recalculation of Moment at Mid Span for Area Model References vi

8 List of Figures Figure 1.1: Torsion in Circular Sections...1 Figure 1.2: Shear Centers of Common Shapes...2 Figure 1.3: Warping Sections...4 Figure 1.4: Torsion in a W Section...5 Figure 1.5: Torsion in Solid Circular Sections...7 Figure 1.6: Torsion in Thin Rectangular Sections...9 Figure 1.7: Twisting of a Rectangular Section..11 Figure 1.8: Thin Walled Hollow Torsional Members..12 Figure 1.9: Torsion in I Girder Sections Figure 1.10: Torsion Moment applied to W12x Figure 2.1: Airline Highway Interchange-Louisiana 22 Figure 2.2: Saint Venant s Torsion...24 Figure 2.3: Warped Section...25 Figure 2.4: V-Load Method...26 Figure 2.5: V-Load Method...27 Figure 2.6: V-Load Method...27 Figure 2.7: FEM model Example Problem..31 Figure 2.8: Girder Cross Section Figure 3.1: Typical Curved I-Girder Bridge..33 Figure 3.2: Construction of Open and Box Girder Bridges...36 Figure 4.1: Tangent Bridge-Approach Span Ironton Ohio...38 Figure 4.2: FEM of a Curved Girder and a Straight Girder...41 Figure 5.1: Cantilever Model. (Straight Girder) 47 Figure 5.2: Cantilever Model. (Curved Girder)...50 Figure 5.3: Shear Flow in I Girder...53 Figure 5.4: Strain Rosette...53 vii

9 Figure 5.5: Strain Rosette on Web to Measure Transverse Shear Stresses..54 Figure 5.6: Normal Stress Distribution I Girder 54 Figure 5.7: Instrumentation for Moment...55 Figure 5.8: Shear Stress Distribution due to torsion...56 Figure 5.9: Strain Rosette...57 Figure 5.10: Strain Rosette on a Girder Cross Section for Torsional Shear Stresses...58 Figure 5.11: Instrumentation Package on a Girder Cross Section...59 Figure 5.12: Modified Instrumentation Plan for Torsional Stresses...60 Figure 5.13: Warping Stress Distribution..61 Figure 5.14: Final Instrumentation Package on a Girder Cross Section Figure 6.1: Secondary Effects in the Curved Girders...64 Figure 6.2: Finite Element Model of the I-girder Bridge...65 Figure 6.3: Lanes Defined on the Bridge...66 Figure 6.4: Shear Stresses due to Truck Load...67 Figure 6.5: Normal Stresses due to Truck Load...67 Figure 6.6: Shear Stresses due to Lane Load...68 Figure 6.7: Normal Stresses due to Lane Load Figure 6.8: 3-D Model Frame Element Girders...69 Figure 6.9: Lanes Defined on the 3-D Model...70 Figure 6.10: Bending Moment Diagrams for HS20 Lane Load...71 Figure 6.11: Shear Diagrams for HS20 Lane Load...72 Figure 6.12: Torsion Moment Diagrams for HS20 Lane Load...72 Figure 6.13: Bending Moment Diagrams for HS20 Truck Load...73 Figure 6.14: Shear Diagrams for HS20 Truck Load...74 Figure 6.15: Torsion Moment Diagrams for HS20 Truck Load...74 Figure 6.16: Bending Moment Diagrams Comparison Radius 300 ft vs Radius 175 ft Figure 6.17: Bending Moment Diagrams Comparison Radius 300 ft vs Radius 175 ft...76 viii

10 Figure 6.18: 3-Girder Model Figure 6.19: Load Applied Figure 6.20: Transverse Shear Stresses I-Girder...80 Figure 7.1 Shear Flow/Stress and Instrumentation of an I-girder for shear...84 Figure 7.2 Normal Stress and Instrumentation of an I-girder for Moment...85 Figure 7.3 Shear Stress and Instrumentation of an I-girder for torsion...86 Figure 7.4 Normal and Shear Stresses in an I-Girder due to moment, shear and torsion...87 Figure 7.5 Internal Stresses in an I-Girder...88 Figure D Representation of the stresses...89 Figure 7.7: 2-D state of stress...92 ix

11 List of Tables Table 1: Recalculated shear, moment and torsion...50 Table 2: Applied shear, moment and torsion...51 Table 3: Recalculated shear, moment and torsion-curved girder...52 x

12 Chapter 1- TORSION 1.1 INTRODUCTION Structural members are often subjected to torsional stresses or torque resulting from the forces and moments about the member axis not passing through the shear centre of the members. As the analysis of a member subjected to torsion, ranges from simple circular sections and advances towards the non circular members and then to open members, the evaluation of stresses gets more involved. Torsion is evident in most unsymmetrical loaded members; examples would be shafts of engines and curved bridges. Torsion in its purest form was investigated by Saint Venant in the 19 th century. Torsion usually acts in tandem with the bending stresses or warping stresses. Warping stresses are usually produced in open sections in which the transverse elements do not remain plane after bending. T T 90 Figure 1.1: Torsion in Circular Sections This article deals with the behavior of circular, rectangular and open sections subjected to torsion. Figure 1.1 shows a typical circular shaft subjected to torsion.

13 1.2 SHEAR CENTER: It is worthwhile to discuss how torsion is developed in the member and the way to prevent torsion from acting on a member, this section primarily deals with the same. The point through which the applied forces pass when these do not cause any torsion in the member is known as the shear center. The shear center for singly symmetric sections lies on the axis of symmetry. For the doubly symmetric sections shear center lies at the intersection of the axes of symmetry which coincides with the center of gravity. For the sections made up of only two plates, the location of the shear centre shear centre is at the intersection of the two plates. Shear Centers for some common shapes are shown in figure 1.2 below. = CG = Shear Center Figure 1.2: Shear Centers of Common Shapes 2

14 Most of the frequently used shapes for structural design are poor when it comes to resisting torsion. It is advisable to make the resulting forces pass through the shear center of the member, however the complex load paths do make such a situation unlikely. Closed sections such as HSS or circular sections behave well to resist torsion. I girders are generally restrained laterally to resist torsion. 1.3 BASICS OF TORSION: Any cross-section that is subjected to torsion will rotate through an angle θ. If the section is non circular and restrained then this is accompanied by warping of the section. In a situation when the warping is completely unrestrained the torque resisted at any section of the member is given by (1.1) where: T t = resisting moment of unrestrained section. G = shear modulus of elasticity. J = torsion constant for the cross section. θ = angle of rotation per unit length. When the ability of the section to warp is restrained, then this resisting torque is given by 3

15 (1.2) where: T W = resisting moment of restrained section. E = modulus of elasticity. C W = warping constant. θ = third derivative of the angle of rotation per unit length. The total torque resisted, is the sum of the two torques above. (1.3) The existence of the second term depends on whether the section has been restrained against warping or not. Figure 1.3 shows a twisted section that is warping. Figure 1.3: Warping of an open I-shaped section. 4

16 Although I-shapes are most common shapes used in practice, they have a very low resistance to torsion. In such a situation the torsional moment can be reduced by ensuring that the rotation of the member itself is less than the rotation at the ends of the member. Then this reduced torsion can be calculated using a torsion stiffness which is given by k. Figure 1.4 shows an I-shape subjected to a rotation θ. (Singer and Pytel 1987) (1.4) where: T = torsional moment. θ = angle of rotation. Figure 1.4: Torsion in an I shaped Section. 5

17 1.4 TORSION IN SOLID CIRCULAR SECTIONS A prismatic bar with a circular cross section subjected to uniform torsional moment will follow Saint Venants theory (Singer and Pytel 1987) of pure torsion. The shearing stresses in this circular bar increase linearly from the center to the section to its circumference. This theory makes the following assumptions. 1. Static and dynamic equilibrium is satisfied. 2. Plane sections before twisting remain plane after twisting. 3. The centroidal axis of the circular section is straight. 4. The material is prismatic, homogenous, and isotropic and obeys Hooke s law. If these conditions are satisfied, the stress in a member subjected to torsion moment is given by: (1.5) where: T = torque. r = radius of the circular section. J = torsional constant for the cross-section.. Figure 1.5 shows a solid circular section subjected to torsion. 6

18 θ Figure 1.5: Torsion in Solid Circular Sections 1.5 TORSION IN RECTANGULAR SECTIONS: One of the assumptions that are made for solid rectangular sections is that plane sections do not remain plane after twisting (i.e the section warps), therefore newer methods for characterizing torsion in rectangular sections need to be investigated. Literature refers to three methods of analysis when rectangular sections are discussed; 1. Saint-Venant s Semi inverse method. 2. Linear Elastic Solution. 3. The Prandtl Elastic-Member (Soap Film) analogy. All these theories deal in detail with the phenomenon of warping of the rectangular sections. They give an estimate of torsion in the member based on the deflection of a corresponding membrane. This analogy works for the narrow cross-sections where the thickness is very small relative to the corresponding height. Hence it is imperative that we divide the discussion of the rectangular members into Rectangular Sections and Thin Rectangular Sections. 7

19 THIN RECTANGULAR SECTIONS: The cross-sections of many machines and parts are made up of thin rectangular sections (Boresi and Schmidt 2003) which are primarily made to carry bending and shear but might also carry some secondary torsion. As was mentioned earlier, the torsion for such narrow sections are based on the deflections of corresponding membranes, therefore a relation between the deflection and the corresponding torsion can be written as: (1.6) where: T = torque. Φ = torsion stress function. If this equation is integrated over the thickness and the height of the member then an expression for the torque and stress in the thin rectangular section can be obtained. This theory is based on the Prandtl Elastic-Membrane (Soap Film) Analogy. (Boresi and Schmidt,2003). Final expression is: (1.7) 8

20 where: (1.8) θ = Angle of rotation. It is important to note that the solution is approximate and it does not satisfy the boundary conditions. The similarity of this equation with the basic equation for torsion is evident. A generic thin rectangular section subjected to torque is shown in the Figure 1.6. h b Figure 1.6: Torsion in Thin Rectangular Sections 9

21 RECTANGULAR SECTIONS: This section deals with the rectangular sections in which the depth of the section and the width are comparable. The torsion stress function is different in this case and is defined by: (1.9) where: Δ 2 = double derivative The torque as defined earlier is given by: (1.10) Integrating this expression as mentioned above, however this time the torsion stress functions being different the expression for J is a little different. The Torque is given by (1.11) where: (1.12) k1 = Constant. This is a standard expression for torsion of rectangular sections and there are tabulated values for k1 for different b and h in the literature. Figure 1.7 shows the twisting of a rectangular section. The membrane analogy allows the stress distribution on any cross section to be determined experimentally. It also allows the stress distribution on thin-walled, open cross sections to be determined by the same theoretical approach that describes the behavior of rectangular sections. 10

22 Using the elastic membrane analogy, any thin-walled cross section can be "stretched out" into a rectangle without having an effect on the stress distribution due to torsion. Figure 1.7: Twisting of a Rectangular Section 1.6 THIN WALLED HOLLOW TORSIONAL MEMBERS. The discussion in the previous sections was about cross-sections which were not hollow but solid in nature. Both circular and the rectangular sections are solid in nature, however when it comes to hollow sections (Boresi and Schmidt, 2003) which are made up of a number of solid sections the shear stresses do not act on the lateral surface of the hollow section ( Figure 1.8). Hence while using the stress function care has to be taken to ensure that the membrane and the stress function have zero slopes over the hollow region. 11

23 Figure 1.8: Thin Walled Hollow Torsional Members Hollow sections must be studied from basics and by considering the shear flow in the section. Since the section is assumed to be thin, the shear flow is constant and the maximum shear stress occurs where the thickness is minimum. For hollow sections the torque is given by: (1.13) where: Q = area enclosed by the mean perimeter of the section. q = shear flow. t = thickness. τ = shear stress. This equation assumes that the section has constant thickness and hence the shear stress stays constant through the wall thickness. This theory is also based on the membrane analogy provided by Prandtl, (1903). 12

24 The relation between torque and twist can be written in the form (1.14) In this case the expression for J is given by: (1.15) where: R ave 3 = mean radius of the cross-section. The results for the hollow section improve as the ration between t and R ave reduces as the shape tends more towards a solid shape. 1.7 TORSION IN OPEN SECTIONS Open sections, like channels and wide flanges sections, tend to warp under torsional loads (Boresi and Schmidt, 2003). If this warping is totally unrestrained then only pure torsion exists in the member. If the section is restrained then additional normal stresses and longitudinal stresses are introduced, which have to be taken into consideration. The stresses due to pure torsion, direct stresses, or warping are related to the θ that is the rotational constant. As was discussed earlier there are two types of stresses, one arising from pure torsion and the other arising from warping. 13

25 Pure Shear Stresses: These are the stresses that are always present in the cross-section whether or not the section is restrained against warping or not. These are in plane shear stresses that vary linearly across the thickness of the section and they act in a direction parallel to the edge of the element (Figure 1.9). The maximum stress is determined by the equation: (1.16) where: τ t = pure torsional stress at the edge of the element. G = shear modulus of elasticity. θ = first derivative of the angle of rotation per unit length. The pure torsional stress is the greatest in the sections with the greatest thickness. Warping shear stress: T Figure 1.9: Torsion in I-girder section. 14

26 When the sections are allowed to warp freely these shear stresses do not develop. When the ends are restrained these warping stresses develop along with the normal stresses. They are given by the equation: (1.17) where: τ W = shear stress at point s due to warping. E = modulus of elasticity. S WS = warping statical moment at point s. T = thickness of the member. = third derivative of the rotational constant. The normal stresses that are developed in the element when it warps are given by: (1.18) where: σ WS = normal stress due to warping torsion. E = modulus of elasticity. W ns = normalized warping constant at point s. = second derivative of. After the stresses from pure torsion, warping and normal have been calculated it is important to check their sense and then add them to get the worst condition. 15

27 1.8 TORSION EXAMPLE. 10 kip 6 10 kip W 12 X 58 Figure 1.10: Torsion Moment applied to W 12 X 58 Torsional properties of W12x58 Flexural Properties of W12x58 J = 2.10 in4 C W = 3570 in6 A = 66.3 in W no = 28.9 in2 I x = 475 in4 S x = 78 in3 t f = 0.64 in t w = 0.36 in S W1 = 46.3 in4 Q f = 18.2 in3 Q w = 43.2 in3 M u = V u =. σ bx = τ bw =. τ bf = Calculate Torsional Stress Tu = Pu X e = 10 kip X 6 in = 60 kip-in. 16

28 Follow appendix B and C from AISC design guide 9, Torsional Analysis of Structural Steel Members, Seaburg and Carter (1997). From the tables, for mid span, ; α = 0.5 From the tables, for support, ; Stress due to pure torsion = Therefore, Flange, Web, = 1.83 ksi. = 1.03 ksi. Stress due to warping = At midspan; = At support; = )] = ksi. )] = ksi. 17

29 Chapter 2-BRIDGES CURVED IN PLAN 2.1 INTRODUCTION: Horizontally curved bridges have been used extensively over the last 50 years. The application of curved girder bridges is largely dominated by the nature of intersections and ramps, and the alignment of the highways. Research in the field of curved girder bridges goes back a long way to the last century. Many researchers modeled the behavior of the curved girder bridges based on the work of Saint-Venant. Many researchers tried to compute the stress in a curved girder which led to the development of approximate theories in the literature. Federal Highway Administration lead much of the research initiative, which led to the development of software, specifications and modern tools which help present day engineers as design aids. U.S Steel contributed to the cause by publishing a report on a simplified method of analysis in Computer programs became the most widely accepted method of analyzing the curved girders; many of these used the slope deflection equations of the Fourier series to find the deflections and the stresses. Bridges were designed using the software and then the results compared to the field tests. Work by Beal and Kissane (1971) confirms that the planar grid analysis developed by Lavelle (1966) is reliable for estimating the deflections and the in plane stresses for the static loads. In 1969 the Federal Highway Administration with the money contributed by 25 states carried out an extensive research program for a detailed analytical and experimental program for studying the behavior of curved girder bridges. A lot of work was done in the coming years in this field including experimental testing, developing methods of analysis, preparing computer programs and simplifying design methods. The final aim of this massive effort was to create a 18

30 comprehensive guide for the design of curved girder bridges for implementation in the AASHTO Specifications for Highway Bridges. 2.2 LITERATURE SURVEY. If we go back in time, the first work published on curved girder behavior is by Saint Venant (1843). However this did not find its way into mainstream research until the latter half of the 20 th century. When the curved girders were started to be used in the 1960 s, extensive research money was invested by the US government. Apart from the United States of America, Japan is yet another nation devoted to the study of the curved girder bridges. Hanshin Highway Public Corporations design codes for steel bridge structures was published in Japan in 1980, similarly in the US, AASHTO guide specifications (1983) were a result of the work put in the latter part of the 19 th century. Zureick et al. (1994) compiled a database comprising of all the work related to the curved girder structures in the past. McManus et al (1969) was the first one to come up with an extensive list of over 200 references related to curved girder work and this paved the way for the future research. In 1969 Federal Highway Administration (FHWA) set up a consortium of university research teams for studying the analysis and design of the horizontally curved bridges. This team included Carnegie Mellon University, The University of Pennsylvania, The University of Rhodes Island, Syracuse University and the University of Maryland, whose efforts led to the development of the first working stress design criteria. All this work pertained to the development of the knowledge of the curved bridges in general. In regards to the instrumentation and experimental analysis of the curved girder bridges many more 19

31 references in literature can be found. As testing full scale bridges on sites is not a very feasible option many tests were conducted using scaled models of the bridges or of the individual components of the bridges. These tests were then used to check the validity of the analytical studies. Culver and Christiano (1969) performed analysis of a curved girder system to validate a computer analysis package. They made a 1/30th model of a curved structure with twin girders and a deck. The model was run without a deck and checked for stresses too. The two span bridge models were made up of Plexiglas; loads were applied using hanging weights which were increased incrementally. Strains and deformations were noted and then compared with the results of the computer program. There was reasonable success obtained with respect to the stresses obtained in the mid span, however it differed when it came to the middle support. It was concluded that the warping had a significant effect on the stresses of the girders and that the curvature caused the difference in behavior between the interior and the exterior girder. As a part of the CURT project, Brennan (1971) performed an extensive set of experimental studies. These studies involved the development of a three dimensional analysis method and a computer program to compare the findings of the scaled tests performed. Around the same time frame as the Japanese were developing their own specifications a wide range of experimentation and research was carried out on the plate and box girder research. This research was completed in 1977 and a majority of the findings can be found in a paper provided by Kitada (1993). Apart from the specifications a lot of other work was also carried out in Japan. Fukumoto and Nishida (1981) tested six curved I-beams. They used the results from these to come up with a theoretical method that predicted the elastic and inelastic large deflection behavior. This theoretical study predicted the forces and the displacements of the girders using the higher order differential equations and energy methods. 20

32 Nakai et al (1983) performed extensive series of tests studying the local bucking behavior of the curved I-girder panels. These panels were loaded so that the effects of bending moment, shear and torsion on the panels could be studied. This study also included the effects of transverse and longitudinal stiffeners on the panels. This study had a lot of conclusions most of which the authors noted were that the critical bending moment decreases as the curvature reduces. The shear tests also led to the development of empirical formula for determining the ultimate shear strength of the panels. Nakai and Yoo (1988) published a book on the analysis and design of the curved steel bridges which goes into great details regarding the ultimate load equations for bending, shear, and the combination of the two. They start by talking about the basic theory of the thin walled sections and then go on to more complex equations discussing the fundamental theory of the curved girders for analyzing the static and the dynamic behavior. The book also deals with the torsional forces as were discussed in this thesis earlier. Daniel Linzell (1999), as a part of his work in fulfillment of doctoral degree did extensive experimental testing on the curved steel bridges. His work included full scale testing on an experimental set up of an I-girder bridge which is the first test like this in the world. His work included original contributions to the field by studying the behavior of the curved girders during erection and using the tubular members as the cross frames. He also optimized the instrumentation for the cross frames. This project was a part of a much bigger project which involved coming up with new rational guidelines for the curved girder bridges. His dissertation includes a much more populated list of references for the experimental determination of the stresses in a curved girder structure. 21

33 Thus it is clear from this discussion that many researchers all over the world have contributed to this cause and tried to gauge the stresses in a curved girder structure. 2.3 ECONOMY & AESTHETICS: Figure 2.1: Airline Highway Interchange-Louisiana One of the fundamental quantities that govern the design of any structure is the economics. Curved bridges are advantageous in one regard as they can follow the roadway alignment and can simplify sub structure requirements, which results in an overall simplified structure. However more effort has to be put into the fabrication and erection costs of a curved girder bridge than a straight girder bridge. The curved girder also requires specialized labor and shop requirements. However the aesthetic value and the simplification of the overall design, make up for these shortcomings. The Airline highway interchange (Figure 2.1), north of the Baton Rouge in Louisiana is one of the many examples where the curved bridge design using steel box girders makes beautiful solution to a practical problem (source: Google images). 22

34 There is no rule of thumb that states whether a curved girder bridges is less or more economical,. However curved girder structures are much more pleasing structural elements and an increasing effort has been made for the structures to be in harmony with the environment and aesthetically pleasing. The initially built bridges showed fatigue problems due to the details of the connections or the improper distribution of load. All these inconstancies lead to the research in the curved girder field. 2.4 METHODS OF ANALYSIS: The basic stresses that exist on in a straight girder exist in the curved girders; the additional effect is the effect of the shear stress due to the torsion. Thus all the methods involved in the curved girder philosophy are to gauge this stress due to torsion. This stress causes the warping of the girder, thus it acts in addition to the normal stresses produced due to axial force and the bending moments. There are multiple methods to study this torsion, Saint Venant torsion and approximate method of analysis are the two ways to do so which are stated below. Saint Venant Torsion: This torsion exists in its purest form when the section is completely unrestrained to warp. Thus in addition to the stresses existing in the section due to axial force and moment, this shear stress has to be added into the section (Figure 2.2). 23

35 Fig: 2.2: Saint Venant s Torsion Source: Curved Girder Workshop Handbook The equation for the calculation of this moment is given by M=Gkθ (2.1) where: G = shear modulus of elasticity. K = torsional constant for the cross section. θ = angle of rotation. The calculation of the member properties and K in this method is rather cumbersome and time consuming. This formulation is a kind of mixed formulation in which Saint Venant (pure torsion) and warping torsion exist simultaneously. Thus the total moment is the summation of the moments caused by the two. 24

36 Fig: 2.3: Warped Section Figure Source: Curved Girder Workshop Handbook It is interesting to note that most of the finite element software s in practice are based on the analysis stiffness method of analysis. In majority of the times the inaccuracy of the results using these software is due to the presence of this warping torsion. After going through the torsion calculation and the section properties there are different cases based on if the section is composite and non-composite. This process is difficult therefore simpler approximate methods are preferred. 25

37 V-Load Method: The approximate methods assume that the torsional warping stresses are proportional to the increment in the torque or the concentration of a load. The most common method used for the analysis of a curved girder bridge is the V-load method. (Curved Girder Workshop Handbook). This theory is based on the static analysis of a beam curved in plan carrying an axial force. Fig:2.4: V-Load Method Figure Source: Curved Girder Workshop Handbook When a small arc of a curved girder bridge is considered it can be seen from Fig.12 that the axial forces are not collinear (due to the curvature effect). This results in another force referred as the radial force in the literature. 26

38 Fig: 2.5: V-Load Method Figure Source: Curved Girder Workshop Handbook The radial force is assumed to act as a distributed load on the girder. The radial distributed force has the same effect on the flange of the girder as in the case of hoop tension. Thus the radial force keeps the flange of the girder in tension. An analogy can be drawn between this effect of the radial force and the hoop tension. Figure 2.6 explains the development of tension in the flanges. Fig: 2.6: V-Load Method 27

39 Figure Source: Curved Girder Workshop Handbook This V-load results in the moments which have to be added or subtracted to the moments from the straight girder analysis. Thus the total moment for which the bridge is designed is the moment obtained from the V-load analysis plus or minus the moments from the straight girder analysis. Earlier curved girders were not used because of the weak mathematical background behind the mechanics involved. Hence efforts were made to simplify the approach. Following were the techniques used. 1. The plane-grid method. 2. The space frame method. One of the reasons of finding references to the V-load method are because of the extensive work of FHWA, however the two theories discussed below are classical theories which talk about how curved girders are treated as straight and then analyzed for the stresses. Following is a summary of how a curved bridge would be analyzed using these methods: 1. The plane-grid method. The structure in this method is modeled in the form of two dimensional grid elements with each node having one translation and two rotational degrees of freedom. The curved beam hence is approximated as a straight member and warping is ignored. Lavelle and Boick (1965) first introduced this method. This work led to the development of various computer programs, CUGAR1 being one of them. The concept behind this is that the curved beams 28

40 loaded normal to their planes in the bridge are replaced by straight prismatic grid elements with equivalent loading. The procedure followed by these computer programs is that the curve is replaced by a series of grid type straight elements representing chords of the curve. Therefore if a curved beam is replaced by a chord that means much smaller span lengths for the bridges or means uneconomical sections as the over hand would be too large, also a long cord might not accurately gauge the curvilinear nature of the bridge. Douglas et al (1972) did an in depth investigation regarding the analysis of a bridge by using the plane grid method and other mathematically intensive methods. The major difference between the two methods used was that the planar grid method ignores the torsion in the member. These investigations lead to the following results: 1. The planar grid method worked well in predicting the deflections and the in plane bending moments. 2. The lateral flange bending stresses were significant; however the plane grid method was not able to capture these as it ignores warping. Hence, the straight girders in a curved bridge are subjected to warping stresses, which are difficult to capture, by the plane grid analogy. The straight girders of the bridge are just a simple solution not to construct a curved girder bridge. Warping and radial stresses do exist in a bridge curved in nature, replacing the curved girders with the straight ones just simplifies the task but does not get rid of the stresses involved. 2. Space Frame Method. Brennan and Mandel (1973) introduced this method for the analysis of open and closed curved members. The curved beams are idealized as straight members and the diaphragms 29

41 and bracing like truss like members that carry only axial loads. The effect of warping is ignored in this method like the planar grid method. The procedure involved is setting up a stiffness matrix for the three dimensional element and then incorporating it in a computer program to study a curved beam. 2.5 EXAMPLE PROBLEM: To illustrate the procedure a curved bridge was analyzed both in the 2-D form and the 3-D form and the moments compared. The software used for the analyses was SAP The bridge under consideration is a curved, two lane highway bridge, with simply supported, composite, plate girder stringers. Diaphragms are spaced at 15 ft measured along the centre line of the bridge. Outer girder is analyzed, therefore the V-load moments are added to the moments from the 2-D model. 2-D model is just a line model of the simply supported bridge, LRFD loads are placed on it and the moments are found for the Strength I combination. A full scaled 3-D model is created and then modeled in SAP

42 Fig: 2.7: FEM Model Example Problem 31

43 Fig: 2.8: Girder Cross Section RESULTS: The Strength I moments were found out to be as follows. Moment for 2-D model: 10,240 Kip-ft. Moment for 3-D model: 10,800 Kip-ft. 32

44 CHAPTER 3-COMPARISION BETWEEN OPEN AND CLOSED BOX CURVED GIRDER BEHAVIOR. 3.1 INTRODUCTION The curved bridges seen today can be categorized into mainly two subtypes: 1. Slab on I-girder bridges. 2. Box girder bridges. In the early stages of curved girder construction most of the bridges were open I- girder bridges, however recently the trend and the research has shifted to box girder construction. This part deals with the salient features of both types of bridges with regards to mechanics, fabrication, construction, and economics. This will provide a basis on the advantages or disadvantages of using a particular type over the other. Figure 3.1 shows a typical curved bridge consisting of steel I-girders. Figure 3.1: Typical Curved I-Girder Bridge. 33

45 3.2 MECHANICS: I-girder bridges: The vertical bending moments to be calculated can be found by eliminating the effects of curvature and analyzing the bridge as a straight girder having a length equal to the arc length of the curved girder bridge. The effects of curvature on the bending moment can be ignored when the girders are concentric and the bearing lines are not skewed more than 10 degrees from the radial and when the arc span length divided by the girder radius ratio is less than 0.06 radians. The shear connectors are based on the forces in the maximum positive and the negative moment regions, their design is also based on the fatigue between the slab and the top flange of the girder. Box girder bridges: The analysis procedure is using the approximated methods like the V-load method (AASHTO Guide Specifications 2003) which can be used for the analysis of open girder bridges also. The effects of curvature on the bending moment maybe ignored when the girders are concentric or the bearing lines are not skewed and also if the arc spans length divided by the girder radius ratio is less than 0.3 radians. In addition to the provisions for the open curved girder bridges the shear connectors would be designed for torsional shear with the vertical bending shear. 34

46 3.3 FABRICATION AND ERECTION: I- girder bridges: Welded I-girders are fabricated from cut curved flanges. Care should be taken to ensure that the girders are stable during the erection phase, the strength of the bolted connections is important and is considered for evaluating the strength and stability of the steel during erection of an open girder bridge. Torsional restraint of the girders is required at each point of time. Instability arises because the internal bracing cannot resist the lateral movement. The best method to prevent instability during the construction phase is to maintain small unbraced lengths. Box girder bridges: Box flanges are usually cut-curved. Erection of box flanges is a complicated task because of the large torsional stiffness of the girders. The cross-frames and diaphragms for the box girders have to be shop fit, field adjustment is very difficult with the box girders because of their large torsional stiffness. Figure 3.2 shows a typical open girder bridge and box girder bridge during construction. Usually if the bridges get past their erection phase then they serve their design life span. The figures are a small example of the complexity of construction for any curved structure. 35

47 Figure 3.2: Construction of I-Girder and Box Girder Bridges 36

48 3.4 ECONOMY: Economy is the driving factor behind any project. There are some advantages and disadvantages towards using curved girder bridges as a whole over the straight girder bridges. Overall economy is achieved because the curved girder bridges follow the same roadway alignment and money can be saved by limiting the additional structures required. The construction and the fabrication of the curved girders are more expensive than the straight girders. The erection of curved box girders is more complicated than the curved open girder bridges. However the box girders being more stiff offer better serviceability than the open girders. Therefore box girders behave better than the open girders structurally but their construction and fabrication cost is more than the open girders. 37

49 Chapter 4 -TANGENT BRIDGES In the earlier part of the nineteenth century, bridges that were curved had straight girders supporting them. This kind of construction was evident in curved viaduct construction. An example of this kind of construction is the Ironton Russell Bridge in Ironton, Ohio, which was constructed in the early nineteen twenties (Figure 4.1). As is shown the bridge is supported by straight girders although the roadway is curved. Short spans, deeper sections, and a large number of supports are characteristics of tangent bridges. This article deals with the salient features of such bridges with regards to torsional, warping, and radial stresses and comparing these with the stresses in curved girder bridges. Figure 4.1: Tangent Bridge-Approach Span Ironton Ohio 38

50 4.1 INTRODUCTION: Earlier the straight girders were used for supporting a curved profile, this created a discontuinity between the girders. A tangent bridge is aesthetically less pleasing and require more substructures like columns and foundations. Curved girders generally have simpler construction details and small depths owing to uniform overhangs and constant girder spacing. Straight girders can only approximate the curvature and hence leads to non-uniform depths and requires non-uniform construction details and shorter spans. The effects of continuity (interaction between adjacent girders) are maintained and are more significant in curved girder construction than in the tangent bridge construction. Due to the efforts by the FHWA efforts were underway for creating a specification for curved girder bridges. This article is an attempt to understand the significant differences when a curved bridge supported by straight girders and a curved bridge supported by curved girders. 39

51 4.2 MECHANICS: Hall (1996) presented a very detailed analysis which is pertinent to this section. He created finite element models of curved and tangent structures having the same span lengths; a generic loading of one kip per feet was applied. The two systems have the same system of cross frames and bracings (Figure 4.2). The results of the analysis and its impact on this research are as follows: 1. In the tangent girder structure both the girders see equal moment and reactions. 2. In the curved configuration the outer girders carry almost all of the moment. 3. In both cases however the total algebraic sum of the moments is the same. 4. The cross frames and the bracings in the tangent structure see no load at all whereas they take considerable moment in the curved configuration. The results can be interpreted easily. The tangent girder structure takes uniform load and hence shows equal moment whereas due to the curved nature in the curved girder structure the load is distributed more towards the outer girders. Thus in a tangent bridge, the straight girders can only approximate the load taken at a particular section of the bridge. Earlier work on these lines suggested the structures to be built with only curved exterior girders and straight interior ones because they were enough to capture the effect of curvature. 40

52 Figure 4.2: FEM of a Curved Girder and a Straight Girder A major difference between the curved structure and the tangent structure is that the cross frames are present almost only for stability in the straight configuration whereas, they are load carrying primary members in curved structure. Cross frames provide restoring torques to the girders resulting in stabilizing the entire system in curved girder bridges. Going back to the discussion of a curved structure supported by straight girders due to the additional stresses on the girders and the cross frames and the absence of sufficient bracing the sections become deeper and the construction details become painful. This leads to a fatigue problem in the structure due to repeated loading. The spans of the structure are smaller avoiding the curvature effect as far as possible; this leads to more supports for the structure. Although the curved girders are more expensive than the straight girders, problem of the substructure cost being more expensive in the latter case makes the curved girder arrangement more economical. 41

53 Considering the basics, a curved girder is subjected to two major stresses at the same time hence its behavior is far more complicated than a tangent girder bridge. Curved girders perform much better than the tangent girders in every respect. Introduction of the box girder configuration has added the benefit of the torsional stiffness to the structure. 42

54 4.3 LRFD SPECIFICATIONS CURVED vs STRAIGHT: To conculde this discussion, a summary of the specifications between curved and straight girders is presented which helps to understand how different the load resistance factor design has been formatted. As an initiative of FHWA, NCHRP started developing the LRFD sepecification for the curved girder bridges in The specifications for the curved girders were included in the AASHTO specifications in the form of a draft in 2006, the straight girder provisions were included since The load factors in the curved girder specification were the same as in the straight girder provisions. The major differences were that of some checks or equations dropped out while designing a straight girder bridge. The statistical model developed for the straight girders maximizes the effects of dead live and dynamic loading. The distribution factor, while writing the straight girder specification is equal to one, this assumption is not true for the curved girder bridges as the curvature of the load is distributed differently. Therefore coming up with the right distribution factors for the moments and the shear capacity of the curved girders is a big task which is accomplished by creating finite element models of bridges using advanced software and noticing the distribution of forces in field and in the models. This gives a fairly accurate judgment of the distribution factors for the curved girder bridges. It is not surprising that the girder distribution factor for straight girder bridges is 1. Now reflecting back to the task of using straight girders for curved bridges it would be very difficult to predict the distribution factors, the distribution is definitely not equal to one because the overall nature is curved and at the same time it is difficult to apply a the theory of curved girder bridges to it. References in literature to the tangent bridges are mainly attributed to 43

55 experience, hence in absence of concrete theory it is difficult to choose any one path for the problem. This leads to uneconomical sections with a large number of supports to make up for the uncertainty. 4.4 SUMMARY: 1. Curved bridges having straight girders are unable to gauge the curvature of the bridge, this leads to larger overhands and deeper sections of the exterior girders. 2. Since instead of a smooth curve, chords are used it leads to shorter spans than usual and discontinuity between the girders. This also leads to larger number of supports than required. The continuity between the girders is lost and detailing becomes important. 3. Although the individual curved girder costs more than the straight girder a tangent bridge costs more than a curved girder bridge because of the additional costs of the substructure units involved. 4. The straight girders are subjected to radial and warping stresses however most of the analysis methods do not gauge them properly hence they cause lateral flange bending stresses. 5. The straight girder is employed for the simplicity. However with the detailing becoming difficult and shorter span lengths there are fatigue problems in the bridge. With a curved girder bridge once the erection is complete, they have been known to serve adequately. 6. It is very difficult to come up with distribution factors for the curved girder bridges; this leads to a difficulty of distribution the loads properly and hence results in overdesigning the sections. An accurate finite element analysis is the only way to predict the distribution of forces and moments. 44

56 7. When compared to the straight girder bridges the curved bridges have construction issues, it is necessary to be careful while setting up the curved girder bridges because the centre of gravity of the girders do not lie on the girder so it is mandatory to take care while handling and setting up. It is not surprising that when less equipment and knowledge was available in the 1960 s curved deck bridges were constructed using the straight girders because it makes it much easier to construct. 45

57 Chapter 5- STRESS DISTRIBUTION AND INSTRUMENTATION 5.1 INTRODUCTION. Moving ahead in the research after knowing the mechanics involved with the curved girder bridges, an attempt was made to understand the stress distribution in the girders of the curved girder bridge and then develop an instrumentation plan. Instrumentation is a technique used by researchers all over the world to find the type of structural forces which are present while testing anything from bolts to T-stubs to bridge elements. Wide variety of instruments such as strain gauges, linear potentiometers, and displacement transducers are used to measure displacements and stresses. After the study of curved girder system, it is realized that the instrumentation plan which works for a straight girder bridge would not necessarily work for a curved girder structure. Once the actual distribution of stresses in a structure is known it is possible to go ahead and come up with an instrumentation plan which can be further optimized to set up the instruments just at the critical locations. 46

58 5.2 STRAIGHT GIRDER ANALYSIS The aim of this document is to develop a procedure to instrument a curved girder bridge and use the instruments to read the structural actions in a girder accurately. It is not only necessary to understand how forces/stresses are distributed in a girder but also develop an instrumentation package to come up with the most critical shear, moment and torsion. SAP 2000 was used to develop models to understand the distribution of the stresses in a curved girder. A start was made using a simple cantilever straight girder and the level of complexity then increased with a curved girder. Cantilever Model (Straight Girder). Figure 5.1: Cantilever Model. (Straight Girder). This is the bridge girder used for the 3-D FEM model previously mentioned in this article while discussing the V-load method and stress distributions. Salient features of this model are that it 47

59 has 9800 area elements, a typical cross section has 20 elements in each of the flanges and 58 elements in the web. The model looks simple to construct and derive values when you look at it, however recalculating the applied shear, moment and torsion to this simple beam became very difficult. Using the pre processor and the templates in SAP 2000 a lot of effort was put into recalculating the applied shear, moment and torsion from the shell stresses with little success. Finally a model was created manually in SAP 2000 by defining a grid and assigning the areas to it. Three load cases were assigned to the model at the free end. 1. Shear 100 kip. 2. Moment 1200 kip-in. 3. Torsion 1200 kip-in. The aim is to recalculate the applied forces, moments from the stresses in the SAP model area elements. Dividing the flanges and the webs into area elements (shells) gives us a stress contour, which helps us visualize the distribution of the stresses in a girder. Following formulas were used to recalculate the applied forces, stresses from the model. 1. Shear. V= τ 12 * da * No of elements in web. where: V = Shear Force Applied (100 kip). τ 12 = Transverse shear stress in each element. da = Cross Section area of each shell element. 48

60 This process looks simple, however it was realized that the user interface and the SAP 2000 preprocessor made some assumptions which made it difficult for this recalculation to yield accurate results. Finally a model made by editing an input file in SAP was successful in accurately recalculating the shear, moment and torsion. 2. Moment. M= (σ11 * da * LA) * No of Elements in the flanges. where: σ11= Normal Stress. da = Cross Section area of each shell element. LA = Lever arm from the neutral axis. M = Moment applied. (1200 kip-in) As the section is not symmetrical about the x- axis the respective lever arms have to be used with respect to the neutral axis. The contribution of the web to the normal stress was ignored. 3. Torsion. T= (τ 12 * da * LA) * No of Elements in where: τ 12 = Transverse shear stress in each element. da = Cross Section area of each shell element. LA = Lever arm from the neutral axis. T = Torsional Moment applied (1200 kip-in). As the section is not symmetrical about the x- axis the respective lever arms have to be used wrt the neutral axis. The contribution of the web to the torsional shear stress was ignored. 49

61 The results obtained after recalculations was as follows, Table 1: Recalculated shear, moment and torsion. Applied Recalculated Shear 100 kip 93.8 kip Moment 1200 kip-in kip-in Torsion 1200 kip-in kip-in This process gave a reasonable amount of confidence in the SAP 2000 models and as a next step these calculations were sought to be repeated with a curved girder model. 5.3CANTILEVER MODEL (CURVED GIRDER) Figure 5.2: Cantilever Model. (Curved Girder). 50

62 Table 2: Applied shear, moment and torsion. At Half the Span Length Shear Moment Torsion Shear 100 kip at free end 100 kip 5157 kip-in 1600 kip-in Moment 1200 kip-in at free end kip-in kip-in Torsion 1200 kip-in at free end kip-in 1200 kip-in After the success with the straight girder, effort was made to reproduce the same results for the curved girder. The difference in this case is that when a force is applied to one end of the cantilever it creates torsion and bending at some other section. The mid length of the beam was chosen to recalculate the applied shear, moment and torsion. Same loads were applied to the free end of the beam. 1. Shear 100 kip. 2. Moment 1200 kip-in. 3. Torsion 1200 kip-in. At the midspan of the beam these actions were calculated. Using the same equations mentioned under straight girders these structural actions were recalculated. 51

63 Table 3: Recalculated shear, moment and torsion-curved girder. At Half the Span Length Shear Moment Torsion Shear 100 kip at free end kip 4740 kip-in kip-in Moment 1200 kip-in at free end kip-in * Torsion 1200 kip-in at free end - * kip-in * = When a moment/torsion of 1200kip-in is applied to the free end of a cantilever, the torsion/moment respectively due to this is in the order of 62.8 kip-in. 5.4 INSTRUMENTATION: After confidence was attained in recalculating the stresses from the curved girder, an instrumentation plan needed to be developed so that the critical stresses can be measured and re calculated. 52

64 Shear: Figure 5.3: Shear Flow in I Girder Shear stress distribution along the web of the I-girder is parabolic. If the flanges are ignored with respect to the shear stress this curve can be defined by ly 2 +my+n. Using a system of three strain rosettes ( ) an equation can be written for this curve and the shear stress found from the basic principles of strain rosettes. x z o y Figure 5.4: Strain Rosette. Using rosette formula, γ xy = 2 * ε oz - (ε x +ε y ) & τ xy = G * γ xy (5.1) 53

65 Using three sets of strain rosettes the parabolic profile of the shear stress can be determined and the corresponding strains found out. A system of strain rosettes is what is needed to measure the shear force in the web of an I-girder. 3 Sets of Strain Rosette s Figure 5.5: Strain Rosette on Web to Measure Transverse Shear Stresses. Moment Moment can be calculated by measuring the normal stress distribution across the cross section of the girder. σ J Figure 5.6: Normal Stress Distribution I Girder. 54

66 Using traditional BDI strain gages the strains picked up near the flanges of the girder can be found and moment calculated from the formula, σ = M*y/I (5.2) where: σ = Stress calculated from the strain measured. y = Distance of fiber from the neutral axis. I = Moment of Inertia. M = Moment. Therefore, the instrumentation package for an I-Girder to measure moment would be as follows, BDI Gages Figure 5.7: Instrumentation for Moment. 55

67 Torsion. In this section, torsion, which is the most important structural action affecting open sections like the built up I- girder will be discussed. The following types of torsional stresses are prevalent in an open I-girder section. Boresi, P.A and Schmidt, J.R.(2003) a. Pure Shear Stress b. Warping Shear Stress c. Warping Normal Stress Figure 5.8: Shear Stress Distribution due to torsion. The case considered is similar to figure 8b since the cantilever beam is restrained at one end and the section is warping. This stress distribution is parabolic and similar to the distribution seen in 56

68 the web in case of shear. Since the 3 strain rosette system worked in the case of shear stress it should also work for the torsion. If the web is ignored with respect to the shear stress this curve can be defined as ly 2 +my+n. Now using a system of three strain rosettes ( ), the equation for the curve can be written and shear stresses found from the basic principles of strain rosettes. x z o y Figure 5.9: Strain Rosette. Using rosette formula, γ xy = 2 * ε oz - (ε x +ε y ) (5.3) τ xy = G * γ xy Using three sets of strain rosettes the parabolic profile of the shear stress can be determined and the corresponding strains found out. A system of three strain rosettes is what is needed to measure the torsion moment in the flange of an I-girder. Following will be the arrangement of the rosettes as seen in a cross section, 57

69 Strain Rosette Figure 5.10: Strain Rosette on a girder cross section for torsional shear stresses. In a deep built up section like this it is observed that, the top flange of the section picks up a large chunk of the torsional moment applied (about 80%), the gages on the bottom flange are not used to their capacity. This issue needs further research for optimizing the gage package, however this comprehensive package of 6 strain rosettes will pick up all the warping shear stresses produced due to torsion. 58

70 Figure 5.11: Instrumentation Package on a Girder Cross Section. Shear, moment and torsion combined, figure 5.11shows the gage package to capture the worst stresses on a curved girder. These are the ideal instrument locations for picking up the stresses/strains produced by shear, moment and torsion. However, quickly a problem is seen with the instrumentation package for torsion. It is very rare that the top of the top flange is available for a gage to be placed on it, usually in a composite section it would be embedded in a slab, in a more simple case the slab could be sitting on it making it virtually impossible to put a gage on top of it. 59

71 In this case it is possible to use the following package, Figure 5.12: Modified Instrumentation Plan for Torsional Stresses. The gages on the bottom of the top flange should read equal and opposite to that on the top flange. However one gage is missing at the centre of the top flange. This can be overcome by assuming that the stress profile is parabolic and the slope is zero at the middle of the flange at the top. 60

72 Slope is 0 at the top Figure 5.13: Warping Stress Distribution. Say that the width of the top flange is b. We can define the parabolic distribution of the stress as ly 2 +my+n. Therefore, τ xy = ly 2 +my+n. At b=0, τ xy=0, therefore, n=0. (A) At b= 0.5b, 61

73 Therefore, lb+m = 0. (B) At b=b, lb 2 +mb (C) Using A, B and C we can find the equation of the curve for τ xy and also τ xy = G * γ xy Pragmatic gage diagram is as follows. Figure 5.14: Final Instrumentation Package on a Girder Cross Section 62

74 Chapter 6- INSTRUMENTATION OF A CURVED GIRDER BRIDGE 6.1 INTRODUCTION: In the previous chapter there was a discussion on how to instrument a cross-section of an I- girder, however the longitudinal location of the gages along the girder length should also be know, in short the cross-section on a particular span where the stresses are maximum should be known. A three girder model is used for this purpose. This will help in studying the behavior of a system of girders. 6.2 BOWING ACTION: Why is the outer girder of a system of curved girders more critical than the rest? This is due to the curvature and the effect that it has on the redistribution of the forces and the moments in the girders. This needs more investigation which is presented as follows. Most of the amplified forces in the girders are due to the secondary effects (Hall,D,H.1996) which can be explained as follows. 63

75 Figure 6.1: Secondary Effects In The Curved Girders. Once a curved girder bridge tends to deform, the flanges tend to have increasing bowing, which amplifies the curvature and in turn the stresses in the two flanges are no longer equal. This also causes the cross frame forces to be different on the compression and tension flanges. If the shape of the I-girder comes back to normal then warping has to be considered. This proves that the structural actions in curved girders are much more complicated than the straight girder bridges and instrumentation is an effective way of knowing actually what's going on. 64

76 6.3 STRESS DISTRIBUTION METHOD I The distribution of the stresses due to the curved nature of a bridge needs to be investigated further. The 3-D finite element model briefly mentioned in chapter 3 will be used as a case study. The bridge under consideration is a curved, two lane highway bridge, with simply supported (70 ft), composite, plate girder stringers. Diaphragms are placed at 15 ft each. A full scaled 3-D model is modeled in SAP Figure 6.2: Finite element model of the I-girder Bridge. 65

77 The HS loading was run on this bridge to observe the distribution of the loads and moments. Two lanes were defined on the bridge for running the trucks on it. Figure 6.3: Lanes defined on the bridge. The critical stresses to be observed would be the shear stresses (S12 in SAP2000) and normal stresses (S11 in SAP2000). Both these results were studied in detail. A set of studies will be presented to find the most critical points on the bridge. 1. HS20 Truck Load. 2. HS20 Lane Load. Shear stresses and normal stresses distribution can be seen due to these loads on the girders. 66

78 1. HS20 Truck Load. Figure 6.4: Shear Stresses due to truck load. Figure 6.5: Normal Stresses due to truck load. 67

79 2. Lane Load. Figure 6.6: Shear Stresses due to lane load. Figure 6.7: Normal Stresses due to lane load. 68

80 6.4 PLOTS AND CONCLUSIONS: The shear, moment and torsion diagrams have been plotted for this bridge as they give a better idea about the distribution of stresses. In order to generate these plots the same model which is seen in the previous section is used, however this time the girders are built using frame elements in the finite element model. It is easier to read moments and forces out of a frame element than a area element. Following are some snap shots of the frame elements model. Just to refresh, this bridge has a span of 90 ft and radius of 300 ft. All the girders are of the same cross section. Figure 6.8: 3-D model frame element girders. 69

81 Figure 6.9: Lanes defined on the 3-D model. 70

82 Kip-ft OUTERMOST GIRDER INNERMOST GIRDER MIDDLE GIRDER Poly. (OUTERMOST GIRDER) Poly. (INNERMOST GIRDER) Poly. (MIDDLE GIRDER) ft -50 Figure 6.10: Bending moment diagrams for HS20 lane load. 71

83 OUTERMOST GIRDER Kip ft INNERMOST GIRDER MIDDLE GIRDER Figure 6.11: Shear diagrams for HS20 lane load Kip-in OUTERMOST GIRDER INNERMOST GIRDER MIDDLE GIRDER Poly. ( OUTERMOST GIRDER) ft Figure 6.12: Torsion Moment diagrams for HS20 lane load. 72

84 It s clear that the outer two girders are subjected to more moment, shear and torsion than the middle girder. While the moment is highest at mid span, shear and torsion are higher towards the ends. It is also noticed that the magnitude of torsion moment is low. The close spacing of the cross frames and the large radius of the bridge are the reasons for the low moments. This case was simple in the following sense, 1. All three girders were of equal cross section. 2. The radius was 300 ft. 3. Only one load case was run, HS 20 lane load. To have a deeper understanding all these parameters can be varied and then studied. Let s start with running HS20 truck load on the same bridge Kip-ft OUTERMOST GIRDER INNERMOST GIRDER MIDDLE GIRDER Poly. (OUTERMOST GIRDER) Poly. (INNERMOST GIRDER) Poly. (MIDDLE GIRDER) ft Figure 6.13: Bending moment diagrams for HS20 truck load. 73

85 OUTERMOST GIRDER Kip INNERMOST GIRDER MIDDLE GIRDER ft Figure 6.14: Shear diagrams for HS20 truck load Kip-in OUTERMOST GIRDER INNERMOST GIRDER MIDDLE GIRDER Poly. (INNERMOST GIRDER) ft Figure 6.15: Torsion Moment diagrams for HS20 truck load. 74

86 Since the two girders on the extreme of the curve are taking up most of the moment, it would be interesting to see what happens when the radius on this is reduced. The bridge being discussed in the previous sections had a radius of 300 ft; another model having a radius of 175 ft was created. The results for HS20 lane load for the girders are as follows Kip-ft INNERMOST GIRDER 300ft RADIUS INNERMOST GIRDER 175ft RADIUS Poly. (INNERMOST GIRDER 300ft RADIUS) Poly. (INNERMOST GIRDER 175ft RADIUS) ft -50 Figure 6.16: Bending Moment Diagrams comparison radius 300 ft vs radius 175 ft. 75

87 Kip-ft OUTERMOST GIRDER 300 ft RADIUS OUTERMOST GIRDER175ft RADIUS Poly. (OUTERMOST GIRDER 300 ft RADIUS) Poly. (OUTERMOST GIRDER175ft RADIUS) ft -50 Figure 6.17: Bending Moment Diagrams comparison radius 300 ft vs radius 175 ft. 76

88 Interpretations: Although this is a bridge with a very large radius and short span a lot of deductions can be made from the SAP finite element model. 1. The shear stresses caused due to shear and torsion on the girders are critical when it comes to a curved bridge. The web of the girders shows a parabolic variation of shear stresses and this is consistent for all the girders. 2. The top flange stresses, increase from the girder which is innermost to the curve to the exterior. The top flange stresses are also max at mid span. 3. It is observed that the normal stresses are the max at the middle of the bridge (near mid span) 4. Normal stresses show an increase from the innermost girder to the outer girder. 5. Cross frames pick up a lot of bending moment due to the redistribution of stresses that is taking place in the system of girders. It should be noted that the bridge has 3 girders with same cross-section. Conclusions: 1. The outer girders are critical, if limited number of instruments are available then instrument the inner-most girder at mid span for moment and one beam depth away from the support for shear and torsion. 2. The cross frames should also be instrumented and rated since they are also primary load carrying members. 77

89 6.5 STRESS DISTRIBUTION-METHOD II THREE GIRDER MODEL-MANUAL: As mentioned earlier a system of girders is required to study the distribution of stresses on the curved girders. A 3-D curved model was modeled using SAP-2000 and used in studying these effects. Figure 6.18: 3-Girder Model. This model looks simple, however from past experiences of using the bridge modeler and the pre-processor in SAP this model was created manually using the input files. It has 40,000 shell elements and 311 frames. The girders are made up of shell elements and they are connected to the slab with the help of rigid links which are frame elements. The girders are connected to each other with the help of cross frames. The procedure to be used in rating a curved bridge would be 78

90 to place a truck on it (known weight), use the instrumentation package suggested in the previous chapter to figure out the strains then stresses in the girder and then try and use stresses to rate the girder or the stringer. The procedure is the same that would be used on a slab on stringer bridge; however a curved girder needed more instruments on one cross section to measure the stresses accurately. Thus there is a need to know the critical sections on the bridge to optimize the instrumentation plan. This bridge is simply supported having a short span, the girders are the same that have been used in the previous chapter. A static load is put on it and the stress distributed is observed. Static load applied is 100 kip of point load at 4 points, this loading is generic and the magnitude is such that it would create considerable stresses to study the behavior of the girders. Figure 6.19: Load Applied. 79

91 Figure 6.20: Transverse Shear Stresses I-Girder. Following observations can be made: As the load was placed at mid span the middle of the girders are stressed the maximum. The shear stresses are parabolic along the web and there is an increase although minor from the interior to the outside girder. Gage Plan: In a optimized system, use one set of the gage plan prescribed in the previous part on the outside girder near mid-span. Presented below is a flowchart summarizing the two alternatives of rating a curved girder bridge system. 80

92 Table1. Procedure to be followed for load rating a curved girder bridge. Modeling Method Start Instrumentation Method 3-D FE Model of curved Girder Bridge Instrument the outermost curved girders near midspan using the instrumentation systems. Simulate static loads/moving loads, equivalent of test trucks loads on the model Load the bridge with the test trucks Find points of maximum stresses Find the strains then the stresses picked up by the exterior girder Instrument the most critically rated members using the instrumentation systems. Rate the girder using these stresses, knowing the dead load and capacity End 81

93 6.6 FUTURE WORK 1. Increase the number of bridges modeled, varying the radius, span etc. 2. Increase the number of girders, change their type. 3. Study similar effects for multi span bridges. 4. Use field data to corroborate results. 82

94 Chapter 7: LOAD RATING OF A CURVED GIRDER. 7.1 INTRODUCTION. Load rating of bridges is a procedure employed by bridge engineers to find if a particular structural member of the bridge can sustain the load applied. In this document, a procedure to instrument a curved girder bridge is explained and presented in the preceding chapters. The next natural step is to load rate the girder. A discussion about the procedure to load rate the I-girder for shear, moment and torsion individually and combined will be presented in this section. The discussion will be limited to allowable stress design. 7.2 ALLOWABLE STRESS LOAD RATING. The general Allowable Stress Rating equation is (7.1) where RF = the rating factor for the live load carrying capacity. C = the capacity of the member (allowable stress). D = the stress due to dead loads. L(I + 1) = the stress due to live load and impact. In this discussion, all the emphasis will be on finding the live load stress since the dead load stress and capacity for the member will be known. All the sub sections in this chapter are dedicated in finding this live load stress and plugging it into equation 7.1. There are three types of stresses which are read from the instrumentation of the curved girders. 83

95 1. Shear stresses due to shear force. 2. Normal stresses due to bending moment. 3. Shear stresses due to torsional moment. Each case is discussed briefly below. 7.3 LOAD RATING FOR SHEAR, MOMENT AND TORSION Shear Figure 7.1 Shear Flow/Stress and Instrumentation of an I-girder for shear. This figure depicts the use of the three strain rosette configuration to measure the parabolic profile of shear strain. Using the strain rosette equations the strain can be converted to shear stress and shear force by using the area of the web. The load rating equation for shear is as shown in equation 7.2. (7.2) where, RF = the rating factor for the live load carrying capacity. C = the capacity of the member (allowable stress). 84

96 σ D = the stress due to dead loads. τ L (I + 1) = the stress due to live load and impact Moment Figure 7.2 Normal Stress and Instrumentation of an I-girder for Moment. This figure depicts the use of two BDI gages to measure the normal stresses caused by moment. The load rating equation for moment is as shown in equation 7.3. (7.3) where, RF = the rating factor for the live load carrying capacity. C = the capacity of the member (allowable stress). σ D = the stress due to dead loads. σ L (I + 1) = the stress due to live load and impact. 85

97 7.3.3 Torsion. Figure 7.3 Shear Stress and Instrumentation of an I-girder for torsion. This figure depicts the use of strain rosettes to measure the parabolic profile of shear stress caused by warping torsion on the flanges of a I- girder. Using the strain rosette equations the strain picked up by the rosettes on the flanges can be converted to shear stress. The load rating equation for torsion is as shown in equation 7.4. (7.4) where, RF = the rating factor for the live load carrying capacity. C = the capacity of the member (allowable stress). σ D = the stress due to dead loads. τ L (I + 1) = the stress due to live load and impact. 86

98 7.4 LOAD RATING FOR ALL THREE STRESSES AT A CROSS-SECTION OF I-GIRDER The previous section describes the procedure to load rate the beam for shear, moment and torsion separately. To find a critical point on this I-girder cross-section is the question which remains to be answered. Once this critical section is investigated, the principal stresses can be found out and then these principal stresses used as the live load for load rating the I-girder. Figure 7.4 Normal and Shear Stresses acting on an I-Girder due to moment, shear and torsion. Shown in figure 7.4 are all the stresses which act on the I-girder. In order to find the critical section and the principal stresses, the internal forces acting on the girder are plotted in the figure 7.5. Before this part is discussed in detail, it should be remembered that the maximum stresses due to shear, moment and torsion act at different points along the longitudinal profile of the 87

99 girder. This discussion is for illustrating how the points of worst stresses could be found on a I- girder cross section subjected to normal, shear and torsional stresses. Figure 7.5 Internal Stresses in an I-Girder. The point shown in figure 7.4 on the I-girder, located at half thickness of the bottom flange is the point on which all the three stresses act. Similarly at half thickness of the top flange a point can be identified having similar stress conditions. Stresses can be studied at multiple points on this I- girder cross-section. For this discussion, the point which is affected by all three stresses is considered which presents as a worst case. If a cubical element is cut out of the I-girder the state of stress in three dimensions will be as follows. 88

100 σ MOMENT τ TORSION τ SHEAR σ MOMENT τ Figure D representation of the stresses. Shown in figure 7.6 are the stresses acting at the point in consideration on the I-girder. This is a 3-dimensional stress configuration with the shear stress in two directions and the normal stresses in one. Normal stresses are shown on the vertical face with shear stresses on the same face. The shear stresses due to torsion act on the top face of the cube. Using the 3-dimensional equations for principal stresses the maximum positive principal stress can be found out. This principal stress can be used in the load rating equation as the live load stress (equation 7.1) to find out the rating factor. The rating factor equation will be slightly modified as follows. (7.6) where, RF = rating factor for the live load carrying capacity. C = capacity of the member (allowable stress). σ D = stress due to dead loads. 89

101 σ PL = maximum positive principal stress. An important observation would be that there could be multiple points on the I-girder cross section, where the stresses can be calculated. At the neutral axis the normal stress is 0, however shear stresses are acting. At the point discussed in this section normal stresses and shear stress cause by moment and torsion respectively are maximum, whereas only a small part of shear stress due to shear force is experienced. Depending upon the locations of these points in the cross-section, the stresses in two directions or three can be found out. Using these stresses, and the equations for principal stresses or the Mohr s circle the maximum positive principal stress can be determined and the same used in the equations for load rating. 7.5 PRINCIPAL STRESSES. To explain the principal stresses which were talked about in the previous section a small discussion about principal stresses and strains is included. This discussion can be found in any mechanics of materials text book, hence this discussion would be brief and would just describe how to find the 2 dimensional principal stresses. The 2-dimensional state of stress is shown in the following figure

102 σ y τ yx τ xy σ x σ x τ xy τ yx σ y Figure 7.8: 2-D state of stress. The maximum and minimum normal stresses called as the principal stresses can be found from the transformation equation for the normal stresses. Let σ x be the normal stress acting on one of the planes of the body as shown in the figure. Looking at the internal forces following equation can be written for the normal stress. (7.7) Using transformation equations the same equation could be represented as shown below. (7.8) Getting back to finding the principal stresses, the maximum and minimum normal stress from equation 7.7 can be found by taking the derivative of σ x with respect to θ and setting it equal to zero. 91

103 (7.9) From the previous step, an equation for the angles at which the principal planes exist can be found out. (7.10) The subscript p indicates that the angle defines the orientation of the principal planes. The two values of θ p should be between degrees and should differ by 90 degrees. After finding the angles of the principal planes these angles can be substituted in equation 7.7 to find the principal stresses. Another way of finding the principal stresses is by drawing a Mohr s circle. The principal stress in 3 dimensions can be found out in a similar way, by differentiating the normal stress. The method used in this case is Lagrange multipliers and can be found in literature. 7.6 SUMMARY LOAD RATING 1. I-girder cross-section can be rated individually for moment, shear and torsion using the equation 7.2, 7.3 and 7.4 mentioned at the beginning of this chapter. It should be remembered that these actions are not critical at the same cross-section. In the single span case discussed in this document, moment is maximum at mid-span of the beam. Shear and torsion are more critical towards the ends of the beam spans. The best place to instrument and load rate the girder for shear and torsional stresses would be at one beam depth away from the support where the shear force and torsion moment are near their maximum values. 2. A good question which could be raised here is to how to load rate the I-girder for shear, moment and torsional stresses acting together on one cross section. This case was discussed in the section 7.4, where figure 7.5 shows all the three stresses. A critical point can be 92

104 identified on the cross-section of the girder. This critical point may have some or all three stresses acting on it. Using mechanics, the principal stresses can be found out at each location that the stresses seem critical. These principal stresses would then be used for load rating equation (equation 7.6) as the live load stress. In section 7.4 an example to show how this can be done is presented. Many such points can be identified and principal stresses found if the points seem critical. 3. In the previous two points a lot of conditions are presented. After all the steps in the previous sections are followed a critical cross section along the length of the girder can be identified and load rated. An experimental study to corroborate these statements could be a project to be undertaken in the future. 93

105 Appendix A- EXAMPLE: DESIGN OF A CURVED STRINGER BRIDGE EXAMPLE: Design of curved stringer bridge. To illustrate the design procedure of a two lane highway bridge with simply supported, composite, plate girder stringers is designed. The stringers are concentric and the support and the diaphragms are placed radially. Outer girder spans 90 ft and has a radius of curvature of 300ft. Spacing of diaphragms along this span is 15 ft. Distance between the inner and outer girders is 22 ft and G3 is halfway between them. There are safety walks spanning 3.33 ft on each side. Structural steel used is Grade 36 and the concrete for the deck is Class A having 28 day strength of 4000 psi. Figure 1 shows the details of the cross section. 3-3 G1 G2 G3 D=22 94

106 L1=90 along G1, L2=88.7 along G2 G1 G2 R1=300 G3 D=22 R2=289 R3=278 L3=83.4 along G3 G1 Figure.1: Three Girder Single Span Curved Girder Bridge Assumption: Girders will not be shored during construction; therefore weight on the steel stringer includes the weight of the concrete slab and the weight of the stringer and the framing details. This problem is solved first by allowable stress design (ASD) and then by load resistance factor designs (LRFD) by hand calculations and 2-D model. Then a 3-D model is created for the same 95

107 and run in SAP-2000 and results compared. The part on the ASD and LRFD is more elaborate and includes a lot of calculations. The procedure followed for the hand calculations is the V-load method which is found in a lot of literature for an approximate analysis of the curved girder. Solution: Load, moments and shears for stringer For V Loads: where R radius of curvature D - Spacing along span = 440 V loads : They act at the diaphragms downward on G 1, upward on G 3 to resist the torque due to curvature. The reaction due to these loads is: R v = = kps Dead load carried by the steel beams :- (kips/ft) 1) Slab: 0.15 = 0.82 kips/ft 2) Haunch : 0.15 x ( ) = 0.15 kips/ft 3) Steel stringer and framing details = 0.3 kips/ft 96

108 DL per stringer = 1.27 kips/ft Similarly stringer G 2 V loads :- For girders 1,2 & 3: 1286 kip-ft kip-ft kip-ft = 3677 kip-ft. V load = = 8.35 Maximum shear at supports for G 1 maximum dead load shear is sum of the preliminary shear and V-load shear. V DL = = 73.4 kips Superimposed Dead Load Initial dead load V-load moments M V1 in G kip-ft (16.25x x x15) M V3 in G kip-ft ( ) same logic Dead load final moments:- M 1 in G = 1767 kip-ft. M 2 in G kip-ft. 97

109 M 3 in G = 659 kip-ft. Dead load carried by composite section, kips/ft Two parapets: x 2 x 0.15 x 1.5( ) = kips/ft Two railings: x 2 x = 0.01 kips/ft Two safety walks:- ( ) x 2 x 0.15[3.25( ) x ] = kips/ft Future wearing surface: ( ) x 22 = kips/ft SDL per stringer:- M pn for SDL kip-ft M p1 in G kip-ft M p2 in G kip-ft M p3 in G kip-ft V loads 3.90 [ ] R v = /2 = 7.57 kips SDL load moments :- (kip-ft) M V1 in G kip-ft 98

110 M V2 in G kip-ft Final moments :- (kip-ft) M 1 in G kip-ft M 2 in G kip-ft M 3 in G kip-ft Live load models are obtained by a SAP model or equivalent finite element software which does live load analysis. Live load moments M LL G kip-ft G kip-ft G kip-ft Centrifugal force Design speed = 30 mph (cause of sharp turn) = 6.68 (30) 2 / 295 = 20.4% 99

111 C = 6.68 (30) 2 / 2.84 = 21.2% [295,284 are the distances to the 2 axles] H =32 x x = kips This causes a torque of 8 x = kip-ft Resisted by a downward vertical force on G 1 and upward vertical force on G 3 :- P = / 22 = 4.83 kips Maximum moment Mc in G1 Mc= = 202 ft kip 1340 kip-ft = maximum moment Mp1 for truck load on lane 1 Similarly, Shear Vc = = 9.7 kips AASHTO specifications require a wind load of minimum 0.1 kip/ft Torque due to wind = 0.1 x 8 = 0.8 kip-ft Downward vertical force of 0.8/22 = kip/ft V WL = ½ x x 90 = 1.63 kip Maximum moment in G1 is M WL = (0.036 x 90 2 )/8 = kip-ft Combined centrifugal force and wind 100

112 V C + V WL = = 11.3 kips Similarly moment:- Mc + M WL = 202 kip-ft + 36 kip-ft Load combinations: - 1.3[DL + 5/3 (LL+I) + CF] bending 1.41[DL + 5/3 (LL+I) + CF] shear Properties of composite section Article 12.4 says lateral bending stress in the flanges due to curvature must be taken into account Girder G 1 is designed Requirements ( ratio 1/25 For girder alone 1/30 Thickness of the stiffened web t w = De / 577 = = in *Determination of shear capacity Art V p = 0.58 fy d t w =0.58 x 36 x = 548 kips Transverse stiffener spacing of 5 ft or 60 in = 60/60 = 1 < 3 OK 101

113 > OK Co-efficient C C = k/ (D/t w ) 2 Fy K = 5 [1 + 1/(d o /D) 2 ] = 10 D/t w = 60/0.438 =137 C = x 10 / x 36 = Shear capacity of girder:- Vu = 548 [ ] = 327 kips 327 > Vmax Section is adequate for shear Flexure Check [G1] Non-composite section Section modulus top of steel 1970 in 3 Bottom of steel 2750 in 3 Long term composite Top of steel 3800 in 3 Bottom of steel 3100 in 3 Top of concrete 2900 in 3 Short term composite Top of steel 7900 in 3 102

114 Bottom of steel 3300 in 3 Top of concrete 5400 in 3 Top of steel Bottom of steel DL 2297 x 12/1970 =13.99 ksi 2297 x 12/2750 =10.02 ksi SDL 1090 x 12/3800 = 3.44 ksi 1090 x 12/3100 = 4.22 ksi LL+I+CF 3102 x 12/7900 = 4.71 ksi 3102 x 12/3300 = ksi L 34 x 12/81 = 5.04 ksi 91 x 12/161.5 = 6.76 ksi Total ksi < 36 ksi OK Total: ksi < 36 ksi OK Deviations from AASHTO 1. HS20-44 LL is imposed instead of the HL-93 loading 2. Impact is 25% instead of 33% 3. The steps followed for centrifugal force and wind are different. 4. Load combinations are different. 5. Strength and service checks not included. 6. Shear capacity calculation defers. LRFD DESIGN : CHECK FOR FLEXURE (STRENGTH LIMIT STATE) LRFD GIRDER G1 103

115 Fy = 36 ksi F c = 4000 psi Check 3.76 P t = 36 x 22 x 2 = 1584 kip P w = 36 x 60 x 7/16 = 945 kip P c = 36 x 18 x 3/2 = 972 kip P s = 0.85 x 4 x 90 x 7.5 = 2295 kip P t + P w + P c > P s Plastic neutral axis is in the top flange. yy = (t c /2) ( ) = (1.5/2)( ) = 0.93 (from top of steel) D cp = 0 Web is compact D p = = 8.43 [Assuming 7.5 is the depth excluding the sacrificial wearing surface] For composite sections in positive flexure 104

116 M u + fl S xt f mn M u = 6489 kip-ft [ASD load combination] D t = = D t = 6.35 D p > 0.1 D t M n = M p [ D p /P t ] Distances from the component forces to the DNA are as follows :- d s = = 4.68 in d w = = in [36.06 distance to the N/A from the steel section] d t = ( ) = in M p = ( ) [u 2 + (t c - u) 2 ] + [P s.d s + P w.d w + P t.d t ] = ( ) [ 2 + ( )] + [2295 x x x 61.93] = = kip-in = kip-ft M n = [ ] 105

117 = kip-ft > 6489 kip-ft OK Check for shear strength limit state V u = 297 kips [ASD calculations] d o = 60 k = 10 = = = 125 > 1.4 C= = V a = C V p V p = 548 kips C V p = 368 kips C V p > V u OK 106

118 Both shear and flexure checks are ok when LRFD strength I limit states are used. LRFD Loads DCI (There is only one section) A = [18 x x 7/ x 2] = in 2 W = (1.15) = lb/ft Deck DC1 7.5 x (11x12) ( ) = lb/ft Haunch: - DC1 Width = 18 Average width = [18 + 2(9) + 18] = 27 W haunch = 150 = lb/ft per girder DC2 W barrier = = lb/ft per girder Wearing surface (DW) W = = 440 lb/ft per girder 107

119 Distribution factors for moment Interior Girder :- One lane loaded eg = A = in 2 K q = 8 ( x ) = in 4 DF = ( ) 0.1 = (0.90) (0.53) (1.14) DF = Interior Girder :- Two or more lanes loaded DF = x 1.14 = x 0.65 x 1.14 = Exterior girder :- One Lane loaded 108

120 Using lever rule DF = = Multiple presence = Exterior girder :- Two or more Lanes loaded e = = DF = x = Max DF =

121 Distribution factor for shear Interior girder One lane loaded DF = = 0.8 Two or more lanes loaded DF = = Exterior Girder DF = Two or more lanes loaded e = = DF = DF for shear =

122 LFRD model and LFRD capacity 2-D The LFRD capacity as found out earlier was M n = kip-ft V n = 368 kips From the spread sheet M u = kip-ft V u = kip M n > M u and V n > V u OK. Following are some figures of the results obtained from the 2-D model. Figures 2 and 3 are the results of the loads applied on the structure. Figures 4 and 5 are the results from applying the strength 1 load combination. 111

123 Kips Figure.2: Shear Force Diagram for 2-D Model Kips-ft Figure.3: Bending Moment Diagram for 2-D Model 112

124 Kips Figure.4: Strength I Shear for 2-D Model Kips-ft Figure.5: Strength I Moment for 2-D Model. 113

125 3-D Model: A full scale 3-D model was made in SAP-2000 using the bridge modeller module. All the girders were modelled using area elements. The finite element model looks like as shown in figure 6. Figure. 6: Layout of the Finite Element Model The layout of the bridge is as shown in the figure 7 and the lines in yellow show the cross diaphgrams. 114

126 Figure.7: Girders and Diaphragms of the Finite Element Model Two lanes are defined on the bridge as shown in figure 8. Figure.8: Lanes on the Finite Element Model. The model was run for all the load cases that were there for the 2-D case and the results obtained as follows. The Strength I moments for the 3-D model came out to be Kip-ft 115

127 Appendix B-RECALCULATION OF SHEAR MOMENT AND TORSION FOR A STRAIGHT BEAM. Shear. V= τ 12 * da * No of elements in web. where: V= Shear Force Applied (100 kip). τ 12 = Transverse shear stress in each element. da= Cross Section area of each shell element. The following table represents the above calculation done using Excel. As there were 58 elements in the web the calculation was done for each of the 58 elements. The Results are as follows Total=93.87 kip. 116

128 Table1: Recalculation of Shear. Element Stress τ12 Thickness Width Shear Force # ksi in in Kip

129 Element Stress τ12 Thickness Width Shear Force # ksi in in Kip

130 Moment. M= (σ 11 * da * LA) * No of Elements in the flanges. where: σ11= Normal Stress. da= Cross Section area of each shell element. LA= Lever arm from the neutral axis. M= Moment applied. (1200 kip-in). The following table represents the above calculation done using Excel. There were 20 elements in each of the flanges, however the calculation differed as the lever distance was different. Total= kip-in. 119

131 Table 2: Top Flange Moment Recalculation. Element S11 Area Lever Arm Moment # ksi in2 in kip-in

132 Table 3: Bottom Flange Moment Recalculation. Element S11 Area Lever Arm Moment # ksi in2 in kip-in

133 Torsion. T= (τ 12 * da * LA) * No of Elements in the flanges. where: τ 12 = Normal Stress. da= Cross Section area of each shell element. LA= Lever arm from the neutral axis. T= Torsional Moment applied (1200 kip-in). The following table represents the above calculation done using Excel. There were 20 elements in each of the flanges, however the calculation differed as the lever distance was different. Total= kip-in. 122

134 Table 4: Top Flange Torsion Recalculation. Element S11 Area Lever Arm Moment # ksi in2 in kip-in

135 Table 5: Bottom Flange Torsion Recalculation. Element S11 Area Lever Arm Moment # ksi in2 in kip-in

136 Appendix C- RECALCULATION OF MOMENT AT MIDSPAN FOR AREA MODEL. This appendix deals with the recalculation of the moment at mid span for the area model (3-D) and comparing to the bending moments in the plots which have been reported in chapter 6 of this document. M= (σ 11 * da * LA) * No of Elements in the flanges. where: σ11 = Normal Stress. da = Cross Section area of each shell element. LA = Lever arm from the neutral axis. The following table represents the above calculation done using Excel. There were 16 elements in each of the flanges, however the calculation differed for the top and bottom flange as the lever arm was different. This calculation is the same as the last appendix, however this time there are two different approaches to the same 3-D finite element model. For convenience a frame model was used to plot the moment, shear and torsion responses in chapter 6 to study the distribution of moments, shears and torsion in a single span model, however since a area model was used validate the instrumentation plan in chapter 5 the values from the frame model need inspection and comparison to an equivalent area model. 125

137 Figure 1: Frame Element Model Area Element Model 126